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For a positive integer $n$ and $0<p<1$, let $D(n,p)$ be the distance from $p\\binom{n}{2}$ to the nearest integer. Our main result is that, for fixed $k\\ge 4$ and for $n$ large, the minimum of $u_k(G,p)$ over $n$-vertex graphs has order of magnitude $\\Theta\\big(\\max\\{D(n,p),"},"verification_status":{"content_addressed":true,"pith_receipt":true,"author_attested":false,"weak_author_claims":0,"strong_author_claims":0,"externally_anchored":false,"storage_verified":false,"citation_signatures":0,"replication_records":0,"graph_snapshot":true,"references_resolved":false,"formal_links_present":false},"canonical_record":{"source":{"id":"1404.1206","kind":"arxiv","version":3},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2014-04-04T10:07:51Z","cross_cats_sorted":[],"title_canon_sha256":"3dd66671d86e7135473cf4fb9b671f433330405ab2a7205b81f74e1b0802be69","abstract_canon_sha256":"e791f97a65d03d5d7051d5baecf9ac9709eb7d45e40bc7cf71d10deed618b13b"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T00:13:48.707559Z","signature_b64":"dA5rcDwrTGeX3dcmVppSJM8pv8IvoAltfwCZ/nCsG3dC4I6sbcP1d4DExPmFukwoZmBLAdrsqopMjhkEQpYyDQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"3006a5512573cde8289c588df7b9ae786280344822efa26d10f85a2d24b5c73a","last_reissued_at":"2026-05-18T00:13:48.707018Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T00:13:48.707018Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"How unproportional must a graph be?","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Alex Scott, Humberto Naves, Oleg Pikhurko","submitted_at":"2014-04-04T10:07:51Z","abstract_excerpt":"Let $u_k(G,p)$ be the maximum over all $k$-vertex graphs $F$ of by how much the number of induced copies of $F$ in $G$ differs from its expectation in the binomial random graph with the same number of vertices as $G$ and with edge probability $p$. 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